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Convection of vortex filaments
Hi David
I would like to ask whether the convection process of EF,PC and PC2B methods only defines the X coordinates of the two ends of the vortex filament, then how are the Y and Z coordinates defined, and are they obtained simply by multiplying the X coordinates by the cosine or sine azimuth Angle? In this case, the vortex point will only move inside the cylinder, but will not expand out. In addition, I also want to ask how the stretching is considered, if YZ is constant, the vortex will only stretch forward and backward along the X direction. Because I see that no matter according to the formula considering the change of the radius of the vortex core with the age Angle, the induced velocity is eventually returned, and the coordinate defined X is also related to the induced velocity. I may have a relatively simple understanding and cannot understand how the vortex point moves and expands along the direction of YZ.
Can you help me to explain how you implement in the code, such as the change of the radius of the vortex core with the age Angle and the calculation of the filament stretch, induced velocity, how these interact with the XYZ nodes at both ends of the filament.
Thank you so much again！
BR，
Roby
Hi Roby,
this sounds like a misunderstanding on your side. The variable X in the formulas stands for the 3-component position vector (and not just the X component). So every vortex node individually is propagated by either EF, PC or PC2B integration. The rate of change (velodity) of this propagatin is evaluated from wind inflow and wake (self-induced) velocities.
The stretching is explicitly included in this, when two vortex nodes move away from each other the connected vortex element is automatically stretched and the core size is adjusted based on the filament strain.
BR,
David
Hi David
Thank you very much for your quick reply.I have a few more questions:
1. if I want to use the PC2B method, formula 3 depends on formulas 1 and 2, right?At each time step, the calculation results based on the first two formulas are obtained.
2.As you mentioned, the X with the arrow represents the position vector. So in formula one I can understand the position vector at time step t+1 in relation to time step t, so what does the arrow tx+1 in formula two and three represent? Is it still a relationship of position vectors at different time steps?Should we get the relationship between Xt and Xt+1 at different time steps?
3.The induced velocity is a vector coincidence, which represents three directions, but what about the incoming velocity V infinity, is it a value in the X direction?
Thank you so much.
BR,
Roby
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Hi Roby,
I see that there is a mistake in the last two equations tx+1 should rather be xt+1.
The predictor corrector scheme essentially evaluates the velocities at multiple locations. From the initial location (xt) and the “predicted” location xt+1. The nodes are then propagated (from their initial x position) with the average velocity evaluated at these two locations. The PC2B scheme also takes into account the positions from previous timesteps (xt-1, xt-2) in the evaluation of the new position.
In general the inflow (wind) velocity can also have three components, depending on the horizontal or vertical inflow angle and the turbulent velocity fluctuations.
BR,
David
Hi David
haha, You may have forgotten that this is the formula you wrote in your doctoral thesis, and it may have been a long time since you graduated. I am reading and translating them carefully, trying to do the code restoration, and I have also read a lot of your papers, which have helped me a lot, thank you very much.
Am I correct in correcting the formula to look like Figure 2?
And by the way, this V infinity, when we do the calculation, does it correspond to three different directions and different vector directions, or does it just represent the Vx direction?
Thank you again,
BR,
Roby
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Hello Roby,
your formulas look correct, except for the last one, which should start with xt+1,PC2B.
All velocities are always evaluated as 3-component vectors.
BR,
David
